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The μ-recursive functions (or partial μ-recursive functions) are partial functions that take finite tuples of natural numbers and return a single natural number. They are the smallest class of partial functions that includes the initial functions and is closed under composition, primitive recursion, and the μ operator.

The smallest class of functions including the initial functions and closed under composition and primitive recursion (i.e. without minimisation) is the class of primitive recursive functions. While all primitive recursive functions are total, this is not true of partial recursive functions; for example, the minimisation of the successor function is undefined. The primitive recursive functions are a subset of the total recursive functions, which are a subset of the partial recursive functions. For example, the Ackermann function can be proven to be total recursive, but not primitive.

Initial or "basic" functions: (In the following the subscripting is per Kleene (1952) p. 219. For more about some of the various symbolisms found in the literature see Symbolism below.)

Constant function: For each natural number n{\displaystyle n\,} and every k{\displaystyle k\,}:

f(x1,…,xk)=n{\displaystyle f(x_{1},\ldots ,x_{k})=n\,}.

Alternative definitions use compositions of the successor function and use a zero function, that always returns zero, in place of the constant function.

Intuitively, minimisation seeks—beginning the search from 0 and proceeding upwards—the smallest argument that causes the function to return zero; if there is no such argument, the search never terminates.

The strong equality operator ≃{\displaystyle \simeq } can be used to compare partial μ-recursive functions. This is defined for all partial functions f and g so that

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In the equivalence of models of computability, a parallel is drawn between Turing machines that do not terminate for certain inputs and an undefined result for that input in the corresponding partial recursive function. The unbounded search operator is not definable by the rules of primitive recursion as those do not provide a mechanism for "infinite loops" (undefined values).

A normal form theorem due to Kleene says that for each k there are primitive recursive functions U(y){\displaystyle U(y)\!} and T(y,e,x1,…,xk){\displaystyle T(y,e,x_{1},\ldots ,x_{k})\!} such that for any μ-recursive function f(x1,…,xk){\displaystyle f(x_{1},\ldots ,x_{k})\!} with k free variables there is an e such that

The number e is called an index or Gödel number for the function f. A consequence of this result is that any μ-recursive function can be defined using a single instance of the μ operator applied to a (total) primitive recursive function.

Minsky (1967) observes (as does Boolos-Burgess-Jeffrey (2002) pp. 94–95) that the U defined above is in essence the μ-recursive equivalent of the universal Turing machine:

To construct U is to write down the definition of a general-recursive function U(n, x) that correctly interprets the number n and computes the appropriate function of x. to construct U directly would involve essentially the same amount of effort, and essentially the same ideas, as we have invested in constructing the universal Turing machine. (italics in original, Minsky (1967) p. 189)

A number of different symbolisms are used in the literature. An advantage to using the symbolism is a derivation of a function by "nesting" of the operators one inside the other is easier to write in a compact form. In the following we will abbreviate the string of parameters x1, ..., xn as x:

Successor function: Kleene uses x' and S for "Successor". As "successor" is considered to be primitive, most texts use the apostrophe as follows:

S(a) = a +1 =def a', where 1 =def 0', 2 =def 0 ' ', etc.

Identity function: Kleene (1952) uses " Uin " to indicate the identity function over the variables xi; B-B-J use the identity function idin over the variables x1 to xn:

Uin( x ) = idin( x ) = xi

e.g. U37 = id37 ( r, s, t, u, v, w, x ) = t

Composition (Substitution) operator: Kleene uses a bold-face Snm (not to be confused with his S for "successor" ! ). The superscript "m" refers to the mth of function "fm", whereas the subscript "n" refers to the nth variable "xn":

If we are given h( x )= g( f1(x), ... , fm(x) )

h(x) = Smn(g, f1, ... , fm )

In a similar manner, but without the sub- and superscripts, B-B-J write: